Inexact linear solvers for control volume discretizations in porous media

Jan M. Nordbotten,Eirik Keilegavlen

doi:10.1007/s10596-014-9453-8

Abstract

We discuss the construction of multi-level inexact linear solvers for control volume discretizations for porous media. The methodology forms a contrast to standard iterative solvers by utilizing an algebraic hierarchy of approximations which preserve the conservative structure of the underlying control volume. Our main result is the generalization of multiscale control volume methods as multi-level inexact linear solvers for conservative discretizations through the design of a particular class of preconditioners. This construction thereby bridges the gap between multiscale approximation and linear solvers. The resulting approximation sequence is referred to as inexact solvers. We seek a conservative solution, in the sense of control-volume discretizations, within a prescribed accuracy. To this end, we give an abstract guaranteed a posteriori error bound relating the accuracy of the linear solver to the underlying discretization. These error bounds are explicitly computable for the grids considered herein. The afore-mentioned hierarchy of conservative approximations can also be considered in the context of multi-level upscaling, and this perspective is highlighted in the text as appropriate. The new construction is supported by numerical examples highlighting the performance of the inexact linear solver realized in both a multi- and two-level context for two- and three-dimensional heterogeneous problems defined on structured and unstructured grids. The numerical examples assess the performance of the approach both as an inexact solver, as well in comparison to standard algebraic multigrid methods.

Highlights

Control volume discretizations are prevailing for flow calculations in geological porous media, including important applications such as oil and gas recovery and geological CO2 storage [6]
Low-order control volume methods adapt well to the strong material heterogeneities of geological media. These heterogeneous structures are commonly described in a level of detail that makes converged numerical simulations infeasible by traditional approaches. This has led to the development of multiscale approximation methods that aim to capture the main features of the solution to a relatively low computational cost [17]
While multiscale approximations lead to promising results on model problems, they have struggled with robustness issues in more general situations [21]

Summary

Introduction

Control volume discretizations are prevailing for flow calculations in geological porous media, including important applications such as oil and gas recovery and geological CO2 storage [6]. Low-order control volume methods adapt well to the strong material heterogeneities of geological media. These heterogeneous structures are commonly described in a level of detail that makes converged numerical simulations infeasible by traditional approaches. This has led to the development of multiscale approximation methods that aim to capture the main features of the solution to a relatively low computational cost [17]. To remedy the robustness issues, both adaptive [30] and iterative [28] approaches have been suggested

Objectives

Methods

Conclusion